ultrafilter characterisation of huge cardinals

A cardinal $\kappa$ is huge iff there is $\lambda>\kappa$ and a $\kappa$-complete normal ultrafilter on $P_{\leq \kappa}(\lambda)$, or, equivalently, on the set of families of subsets of $\lambda$ of order-type $\kappa$.  I want to know whether one can further assume that $U$ is concentrated on something smaller? For example, can one further assume that $U$ is concentrated on $<\kappa$-directed families of sets of size $\kappa$? Only on those directed families that contain $P_{\kappa}(\lambda)$? Covering families as above ?

In notation: Let $U$ be such an ultrafilter. Let $D_\kappa=\{X\in P_{\leq\kappa}(\lambda): \forall S\subseteq X(|S|<\kappa\implies \cup S \in X\}$ be the subset of all $\kappa$-directed subsets of $P_{\leq \kappa}(\lambda)$. Is $D_\kappa\in U$ ? Let $D'_\kappa=\{X\in D_\kappa: P_{<\kappa}(\lambda)\subseteq X\}$ be the subset of all $\kappa$-directed subsets of $P_{\leq \kappa}(\lambda)$ containing all sets of size less than $\kappa$. Let $C'_\kappa=\{X\in D_\kappa: \forall S\in P_{\leq\kappa}(\lambda)\exists x\in X(S\subseteq x)\}$.
Is $D'_\kappa\in U$ ? Is $C'_\kappa\in U$? Can one find such an $U$ that the answers are positive?

I am also looking for any references describing in detail ultrafilter characterizations of large cardinals.

 The Ultrafilter Characterization of Huge Cardinals Robert J. Mignone Proceedings of the American Mathematical Society Vol. 90, No. 4 (Apr., 1984), pp. 585-590 http://www.ams.org/proc/1984-090-04/S0002-9939-1984-0733411-6/S0002-9939-1984-0733411-6.pdf

The hugeness of $\kappa$ is witnessed by an embedding $j:V\to M$ for which $M^\lambda\subset M$, where $\lambda=j(\kappa)$. In particular, for such an embedding we have $j''\lambda\in M$, and one may accordingly consider the induced measure on $P_{\leq\kappa}(\lambda)$ defined by $X\in U\iff j''\lambda\in j(X)$. The ultrapower by $U$ is a factor of the original embedding $j$ and also witnesses the hugeness of $\kappa$.

Since $j''\lambda$ is $\lt\lambda$-closed, in the sense that every subset of it of size less than $\lambda$ is bounded in it, then we see that $U$ concentrates on such kind of objects. This kind of reasoning is usually the key for understanding the essential nature of the sets on which a hugeness measure must concentrate.

But I am confused about your question concerning $\kappa$-directedness, since it seems to me that every subset $\sigma\subset\lambda$ with order type $\kappa$ will be $\kappa$-directed, since any small subset of $\sigma$ will be bounded in $\sigma$. Have you asked the question you intended to ask?

In your comment you state another property, which I would call something like $\lt\kappa$-closure rather than $\kappa$-directedness, since it refers to the least upper bound $\cup S$ rather than merely to an upper bound of $S$. In this case, $j''\lambda$ is not $\lt\lambda$-closed, since the union of the first $\kappa$ many elements of it is $\kappa$ itself, which is not an element of $j''\lambda$. So the usual hugeness measures will not concentrate on $D_\kappa$ defined that way.

• Thank you! I meant to ask: Is $D_\kappa=\{X\in P_{\leq\kappa}(\lambda): \forall S\subseteq X(|S|<\kappa\implies \cup S \in X\}$ in $U$ ? That is, I am looking at directed families of subsets, not families of directed subsets. – mmm Jul 18 '13 at 15:53
• Maybe I'm misunderstanding something, but it seems to me that, to see that $U$ concentrates on $<\kappa$-closed things, we need to use that $j''\lambda$ is $<j(\kappa)$-closed, i.e., $\lambda$-closed (which it is, of course), not merely that $j''\lambda$ is $<\kappa$-closed. – Andreas Blass Jul 18 '13 at 17:56
• Andreas, you are totally right. – Joel David Hamkins Jul 18 '13 at 18:09
• mmm, what you write is not $\kappa$-directedness, but some kind of $\kappa$-completeness, since you ask for $\cup S\in X$ rather than a cover of $\cup S$. Since $j''\lambda$ does not have $j$ of your property, the usual hugeness measures do not concentrate on $D_\kappa$. – Joel David Hamkins Jul 18 '13 at 18:13
• Thanks! And what about $\beta$-completeness? That is, $D''_\beta=\{X \in P_{\leq \beta}(\lambda): \forall S\subseteq X(|S|<\beta\implies\exists y\in X(\cup S\subseteq y))$, for some $\beta\leq\kappa$ ? – mmm Jul 18 '13 at 18:19